Rearranging Formula | CIE IGCSE Maths: Core Revision Notes 2023

Simple Rearranging

A formula (plural, formulae) is a mathematical relationship consisting of variables, constants and an equals sign
You will come across many formulae in your IGCSE course, including
- the formulae for areas and volumes of shapes
- equations of lines and curves
- the relationship between speed, distance and time
Some examples of formulae you should be familiar with are
- The equation of a straight line
  - $y = m x + c$
- The area of a trapezium
  - $Area = \frac{(a + b) h}{2}$
- Pythagoras' theorem
  - $a^{2} + b^{2} = c^{2}$
You will also be expected to rearrange formulae that you are not familiar with

Rearranging formulae can also be called changing the subject
- The subject is the variable (letter) that you want to find out, or get on its own on one side of the formula
The method for changing the subject is the same as the method used for solving linear equations
- STEP 1
  Remove any fractions or brackets
  - Remove fractions by multiplying both sides by anything on the denominator
  - Expand any brackets only if it helps to release the variable, if not it may be easier to leave the bracket there
- STEP 2
  Carry out inverse operations to isolate the variable you are trying to make the subject
  - This works in the same way as with linear equations, however you will create expressions rather than carry out calculations
- For example, to rearrange $A = \frac{(a + b) h}{2}$ so that $h$ is the subject
  - Multiply by 2

$2 A = (a + b) h$

- - Expanding the bracket will not help here as we would end up with the subject appearing twice, so instead divide by the whole expression $(a + b)$

$\frac{2 A}{a + b} = h$

$h = \frac{2 A}{a + b}$

If the formula contains a power of n, use the nth root to reverse this operation
- For example to make $x$ the subject of $y = a x^{5}$
  - Divide both sides by $a$ first

$\frac{y}{a} = x^{5}$

$\sqrt[5]{\frac{y}{a}} = x$

If n is even then there will be two answers: a positive and a negative
- For example if $y = x^{2}$ then $x = \pm \sqrt{y}$
If the formula contains an nth root, reverse this operation by raising both sides to the power of n
- For example to make $a$ the subject of $m = \sqrt[3]{2 a b}$
  - Raise both sides to the power of 3 first

$m^{3} = 2 a b$

$a = \frac{m^{3}}{2 b}$

If you are unsure about the order in which you would carry out the inverse operations, try substituting numbers in and reverse the order that you would carry out the substitution

Make $x$ the subject of the following formulae.

(a)

4 m + 5 x = 3

(b)

\begin{matrix}  \end{matrix}

3 t = \frac{2}{x}

(c)

A = 4 π x^{2}

(a)

Get 5x on its own by subtracting 4m from both sides

5 x = 3 - 4 m

Get x on its own by dividing both sides by 5

x = \frac{3 - 4 m}{5}

(b)

Remove fractions by multiplying both sides by x

3 t x = 2

Get x on its own by dividing both sides by 3t

x = \frac{2}{3 t}

(c)

Get x² on its own by dividing both sides by 4π

\frac{A}{4 π} = x^{2}

Get x on its own by square-rooting both sides
Use ± to show that there are two possible answers when square-rooting

x = \pm \sqrt{\frac{A}{4 π}}